Laguerre- Hahn orthogonal polynornials with respect to the Hahn operator: Fourth-Order Difference Equation for the rth Associated and the Laguerre- Freud Equations for the Recurrence coefficients

No

d'Ordre:

00008/98

THESE PRESENTEE POUR OBTENIR LE GRADE

DE

DOCTEUR EN SCIENCES

DE

L'UNIVERSITE NATIONALE DU BENIN

Option:

Physique Mathématique

par

Marna FOUPOUAONIGNI

Laguerre· Hahn orthogonal polynornials with respect

to the Hahn operator: Fourth· Order Difference Equation

for the rth Associated and the Laguerre· Freud Equations

for the Recurrence coefficients,

Soutenue le

16

Décembre

1998

devant

le

JURY:

Pr45ident:

Augustin BANYAGA

(Penm,yJvania State University, Penm,yfvanie USA)

Rapporteurs

Saïd BELMEHOI

(Université des Sciences et Technologies de Flandres, Lillt> 1)

M.

Norbert HOUNKONNOU

(Uni~ersité Nationale du Bénin, Bénin)

W. KOEPF

(Hochschule fur Technik Wirtschaft und Kultur, Leipzig, Allemangnc)

A. RONVEAUX

(Facultés U.liversitaires Notre Dame de la Paix Namur-Belgique)

~)(8minateurs

Jean - Pierre EZI N

(Université Nationale du Bénin, Bénin)

Côme GO U DJ 0

(Univer5ilé Nationale du Bénin,

Bénin)

Co~_~i'!~~~~~

M. Norbert HOUNKONNOU

Operator: Fourth-order Difference Equation for the l'th

Associated and the Laguerre-Freud Equations for the Recurrence

Coefficients

:\latna FOUPOUAG:'\IG:'\I

Institut de :\lathéntatiqllcs ct de Sciences

Ph~siqllcS

A rnon épouse AdjonI ct III(/. fille Samillm.

A 1110 fllTl/ille. '!/les IImis el li (,ous eelJ:r qui cl'oient cl. ["cffrnt ct œU:/'Tcnt po Ill' III Justicc. lu, ]llli}: ct 10 dignité l!1lmaine,

Tout d'abord, je remercie le Professeur Augustin BANYAGA qui me fait un grand honneur en ac-ceptant de présider le jury de cette thèse. Je remercie également les Professeurs Saïd BEL:'-IEHDI, J(~an-Piem~EZI:\". Wolfram KOEPF et Côme GOliDJO pour ayoir accepté de faire partie du jury.

J'exprime ici ma reconnaissance aux professeurs ?--1. :'\orbert HOUl\r\:O~-:\"Oliet André RO;'\TEAl:X pour les efforts fournis et les sacrifices énormes consentis pour la co-direction de cette thèse.

yIes remerciements 'ont aussi à l'endroit du Professeur Jean-Pierre Ezin, Directeur de l'EvISP, pour sa constante soli ici t ude.

Je remercie profondément le Serùce Allemand d'Echanges Cniyersitaires (DAAD) qui, en m'octroyant une bourse doctorale, a rendu possible la réalisation de ce tra'·ail. \Ies remerciements "ont aussiàl'endroit de l'Uni,'ersité :\"ationale du Bénin et cn paniculier de l'Institut de :\Iathématiques et de

Sciences Physiqups pOllr l'hospitalité, le somien financier et les sacrifices consentis tout au long de ma formation.

Le sl~jour ('Il Europe. de s('ptenJm' FJ!)-;- ?J mars 1(J98. a <-;t (: déterminant pour la finalisation de CP tra,·ôil. Pour cela. je tiens il remercier le D_·L.\D pour 3yoir fim!llcé ce S(;jour, le Centre

''l'''::onrad-Zusp-ZentnnIl für Infortnationstechnik Bterlin" (ZIB) pour m'a"oir acceuilli en son sein et aussi pour m'a"oir offert d'excellelltPs conàitions de trayail. C'est l'occa::,ion ici de remercier le Professeur \Volfralll h:OEPF pour son hospitalité pendant mon s';jour à I3(~rlin et aussi pour la formation qu'il m'a donnée en calcul symbolique.

Harald B()I:-JG et le Professeur \\ïnfrieè :'\E\\-:\". tous de ZIB. m'ont aussi aidé notalllnlCllt pour la programmation et je le:: en remercie.

Il Ille ]liait ici de remercier tri':: sillci'reI:lCnt le Profe::s(°1lr André~ RO:\"VEAlJX qui, malgré son ad-mission ù la rctraitc, a financ(; à se~ lJlOlJH'S frai" lllOII séjour il "amur et anssi son séjour à Berlin. Ces rapproc!lcmelll sont pe:'mis nnc éyo: nt ioti tr;Js sig,nifîcat in' du t r,n,lil.

.Je remercie <~balemcntG<'rard L.-\G:\JACO pour rachemillP:llcllt des copies de cette thèse à :\"amur . .JOhll TITAI\TAII pour la lecture d:; lllanus, rit ,kCl: trén'lil. h',:ll AHEA de l'l-ni"ersité de \ïgo en Es-pagne, 1<' Professeur Francisco :\IARCELL\:\" d<' 1TLin'r."idad Carlos IIIck?--Iadrid en Espagne pour les discussions fruCTucnscs que nous aY<lllS eucs. le Prufe"Slur ..\ntu:lio J. DCRA:\" de l'l'niversité de S(:yjl!e ('ll Espagnc pour ;l\-oir :l.n,lllcé part i·'llenH'll1 !ll')!. _"{'je.nr à S"v:ll·· et, 1(' pru:essellr \\'alter VA:\" ASSCHE de l'l;niversit<~ de LCU"'JIl en Bclg,i<jl~('[Jour

r

.In'ild\il): .. i'llo-pit;l:il(~~. !'ellG\c:n'llll'nt(JI le suppon financier

dOllt j'ai h{~lll'fi('i<',,,pendant mon sé.:,)1)r ;\ L('l\('l~.

:\1<'s rcnH'n:ieIllcnts vont aussi ;\ !'endr,~it l:e ILOt! pi'l" :\hussa :\10\\'Ol':\L ma mère I\:cntouma :\lAPIE:\JFOC. Illon épouse Adjara FOliPOL\G:\"IG:\"L ma fillc Salllihra FOl-POCAG:\"IG:\"l. mc" fri'lcs :\LJllluuda :\".Jt'TAP:\I\'OCl. :\lallla :;.10YA ..h·:ditldld '\G;\J\:EL Issofa :\IOl'l\'Dl. Ousmanou P..\TOCOSSA ...\hdou :'\GOlîIOUO .... llH" so·urs. ;)(,[lr l,,, sa:rifice" qu'ib ont coIl5entis ;Jendanr mon ahsellu' e\ illissi pour leur soutien i:l<,()ll,litic lllle:.

J'exprinll' aussi m;~ reconnaissa:lce aux -:nsc::gna:lts r~es DéjJ;[rtemClltS de :\lathématiques de l'Ecole '\ormale Snpl;rieure d(- Yaoundé ct dl: lTni-:ersi:é dr: Y;rou:rdé1. au Prof(~sscur:\lu-ise K\\-ATO :\".JOCK pour m';l\'(Jir ('nC(JIlrar:,é à nw pr{~s(':lt('rau (OIlC',1lrS (l'ddlll:S~iO:I h l'I:\ISP ct aussi ]Jour leur suutien er l('urs cous, ,ils.

:\1011S("jolll ;tIT\ISPilfailli Pl('r.riJ" fill (~:_ 19'J:).(('Ci;l C;:l~'~ ':(~Sdifiicnlr/'s a<!millisrratin:s rencontrées ;rn :\Jiliisti'l(' d(·rEduG~tioll '\atioIl,:1r' dll Cc.:I](·r(,ll1l. C":" pI{)Lli:nr'" OIlt <'-tt: l(;s(,lns :ll:UrCnSUIlcuT ct ponl <:<'Ia je rr'l1H'I('](' sillc(~r"l[ICllt les p[l~f<..,sr'llr~ .l';l:l Pi'nf' EZl:\". :\LJh(JlllOli '\mhcn HOU:\"KO:\"'\OC ct :\loïse K \\'A1'0 0.'.J ocr...:: P(jUl leur cOlIlpr<"llf-mivl (.\ lnr Sl;Pl;O:r. pellriallt 1'('\\1:p<'~r iud(~ diF-icill~. Le

sou-tien dn Profl'ss('llr Alp~HllISt';ELOI'\G Pl OYis"ur du L:.\ .;(: (L :,.L:1H:n)';ou1.a<\1' 0.'kongsamba 'J.n Call1<:rOllIl ainsi flnc\('5 inll'ITellti(~.lIsdl'Y(J\[SSvlr'\CJL~,HE('\ :\!èI:i(: U)LISEdn :\lini..;ti~rr: rlr~l'Educatioll

:\"ari'Jlla!r-3

du Cameroun ont été aussi importantes pour la continuation de mes ét.udesà l'IMSP et je les en remercie. Je remercie enfin le personnel enseignant et administratif de nMSP ainsi que les étudiants pour le bon climat de collaboration et de travail, sans oublier tous CCliX qui de près ou de loin ont contribué à

1 Introduction

1.1 Historical introduction . . . . 1.1.1 The fourth-order differential and difference equation 1.1.2 The non-linear difference equations .

1.2 Summary of the main results . 1.2.1 The fourth-order difference equation 1.2.2 The non-linear rccurrence equations 1.3 Outline of dissertation . . . .

2 Preliminaries

2.0.1 The notion of topology . 2.0.2 Notations . 2.1 Orthogonality and quasi-orthogonality

2.1.1 Orthogonal polynomials . . . 2.1.2 Quasi-orthogonal polynomials 2.1.3 Other definitions . . . . 2.1.4 Dual basis . . . . 2.2 Associated orthogonal polynomials . 2.2.1 Three-term recurrence relation

2.2.2 The first associated orthogonal polynomials 2.2.3 The rth associated orthogonal polynomials 2.3 OperatorsD,

Tw,

D""

9

q and Dq .

2.3.1 OperatorD .

2.3.2 Class of the D-semi-classical linear functional

2.3.3 Characterisation of D-classical orthogonal polynomials 2.3.4 Operators

Tw

and D", ..

2.3.5 Class of the Dw-semi-classical linear functional . . . . 2.3.6 Characterisation of 6-classical orthogonal polynomials . 2.3.7 Operators

9

q andDq .

2.3.8 Class of the Dq-semi-classicallinear functional . 2.3.9 Characterisation of Dq-classical orthogonal polynomials 2.4 The q-integration . . . .

2.4.1 The q-integration on the interval [0,a], a> 0 . 2.4.2 The q-integration on the interval [a,0], a

<

0 . 2.4.3 The q-integration on the interval [a, (xl(, a

>

0 . 2.4.4 The q-integrationon the interval ] - 00,a]. a

<

0

3 The Dq,w-semi-classical orthogonal polynomials 3.1 I n t r o d u c t i o n . . . .

3.1.1 Operators Aq,w and Dq,w .

3.1.2 Class of the Dq,w-semi-classicallinear functional 3.2 Characterisation theorems for Dq,w-semi-classical OP .

3.2.1 Dq,w-classical orthogonal polynomials . . . . .

4 7 S 10 10 11 11 13 13 14 14 14 16 16 17 lS lS 19 19 20 20 21 22 23 25 26 27 28 29 31 31 31 31 32 33 33 33 38 41 41

CONTENTS

3.2.2 Dq,w-serni-classical orthogonal polynomials

5

48

4 The formaI Stieltjes function 51

4.1 The Stieltjes fnnction and the Riccati clifference equation 51 4.1.1 Sorne definitions . . . 51 4.1.2 Sorne properties . . . 52 4.2 Dq,...,-Laguerre-Hahn OP as Dq-Laguerre-Hahn OP 57 5 Difference equations for the first associated OP 59 .5.1 I n t r o d u c t i o n . . . 59 .').2 q-classical weight . . . 59 5.3 Fourth-order q-difference equation for P~~1(x:q) 60

.5.4 A p p l i c a t i o n s . . . 61 5.4.1 The first associated Little and Big q-Jacobi polynomials 61 5.4.2 The first associated D-classical orthogonal polynornials . 62 5.4.3 The first associated Dq,w-classical orthogonal polynornials 62 5.4.4 The first associated ~-classicalorthogonal polynomials 63 6 Difference equations for the rth associated OP 65 6.1 I n t r o d u c t i o n . . . 65 6.2 The associated Dq-Laguerre-Hahn linear funcrional . . . 65

6.2.1 The associa ted Dq-Laguerre- Hahn linear funct ional is a D ,,-LaguC'rrc-Hahn lincar functional . . . 65

6.3 Fourth-order difference equation 68

6.3.1 Fourth-order differcntial equation for PAr) 74 6.3.2 Fourth-order diffcrence equation for t.he rth associateci Dq.w-Lagllcrre-Hahn

orthog-onal polynornials . . . 75 6.3.3 Fourth-order difference equat.ion for t.he Tth associatcd LJ.-Lagucrrl2-Hahn

orthogo-nal polynomials . . . 76 6.4 Application of difference equat.ions 1.0 classical situations. . . 76 6.4.1 CoefficientsEr, Fr and Hr for classical si r uations . . . 76

6.4.2 Results on general associated Dq-classical orthogonal polynomials . 77 6.4.3 Fourt.h-order differential equat.ion for the rth associated D-classical ort.hogonal pol

y-nomials . . . 78 6.4.4 Fourth-order difference equation for the rt h associated LJ.-classical orthogonal

poly-nomials . . . 78 7 Three-term recurrence relation coefficients

7.1 Introduction .

7.2 Three-terrn recurrence relation coefficients for Dq-classical situations 7.2.1 CoefficientsTn,1 and T n,2 • . • . • . . . .

7.2.2 Coefficients f3n and -(n for Dq-classical orthogonal polynomials 7.3 Three-term recurrence relation coefficients for D-classical situations.

7.3.1 Coefficients Tn,1 and

T

n,2 " . • . . . • .

7.4 Three-term recurrence relation coefficients for LJ.-classical situations. 8 Laguerre-Freud equations for semi-classical OP of class 1

S.1 Introduction... . S.2 Starting the Laguerre-Freud equations S.3 Interrnediate coefficients

8.3.1 CoefficientsTn,j

8.3.2 Coefficients B~ . 8.3.3 Structure relations

S.4 Final form of the Laguerre-Freud equations

8.4.1 Laguerre-Freud equations for Dq-classical orthogonal polynomials .

81 81 81 81 82 82 82 83 84 84 85 86 86 87 87 89 90

8.5 Applications to D, Dw and Dq,w-semi-c1assical OP of c1ass 1 . . . 91

8.5.1 Laguerre-Freud equations for D-semi-classical OP of class 1 . 91 8.5.2 Laguerre-Freud equations for Dw-semi-classical OP of class 1 91 8.5.3 Laguerre-Freud equations for D w-c1assical orthogonal polynomials 92 8.6 Applications to generalised Charlier and generalised Meixner polynomials of c1ass one 93 8.6.1 Laguerre-Freud equations for the generalised Meixner polynomial of class one 93 8.6.2 Laguerre-Freud equations for generalised Charlier polynomial of c1ass one 94 8.6.3 Asymptotic behaviour . . . 95 9 Conclusion and perspectives

9.1 Conclusion . 9.2 Perspectives.

10 Appendices

10.1 Appendix l .

10.1.1 About D-classical orthogonal polynomials . 10.1.2 About 6.-c1assical orthogonal polynomials 10.1.3 About q polynomials . 10.2 Appendix II . . . .

10.2.1 Results on general associated c1assical discrete polynomials 10.3 Appendix III . . . . 96 96 97 99 99 99 99 100 100 100 105

Chapter

1

Introduction

1.1

Historical introduction

1.1.1

The fourth-order differential and difference equation

Consider the family of monic polynomials {Pn}nEN, orthogonal with respect 1.0 a linear functional 12 (see (2.5)). 11. satisfies a three-term recurrence relation (which we denote TTRR) [Chihara, 1978]

{

Pn+l(X)

=

(x - _(3n)Pn(x) - ~rnPn~l(X), n ~ 1,

Po (x)

=

1,Pl(x) = x - (30,

\vhere (3n and ln are complex numbers \Vith ln

::f

0 'in E~.

The rth associated of{Pn}nEN is the family of monic pol~·nomials {p~r)}nEN,defined by the previous relation in which (3", ln and Pn are replaced by (3n~r, In+r and p~r),respectively,

{

P';~l

(.1:)

=

(.1" - 3n+r )

p,~r)(I)

- rn+rPT;':l1(x),

p~r (x) = 1.P₁(1)(x) = X - (3r.

n ~ 1,

The rth associated of the regular linear functional 12 is, by Favard Theorem [Favard, 1935], the unique line;1.r functional Ôr) \vith respect to ,,·hich {p~r)}nENis orthogonal and satisfices (Ôr1 ,1)

=

Ir.

Let {Pn}nEN be a famih· of polynomials. orthogonal with respect 1.0 the lincar functional12and S(L),

the Stieltjes function of L given by

~ !lIn

S(L)(x)

=

S(x)

= -

L xn~l' ,,>0

where Mn is the moment of order11 ofL: !lIn

=

(12,x n ).

\Vhen S satisfies a Riccat.i differential equation

cP(x)S(x)' = B(x)S(X)2

+

A(x)S(x)

+

D(x),

where cP, A, Band D are polynomials, then {Pn}nEN are called Laguerre-Hahn orthogonal polynomials [Magnus, 1984], [Ozoumba, 1985]. 1t is well-known [:'v1agnus, 1984] that these polynomials satisfy a fourth-order linear differential equation.

Classical and semi-cldssical (continuous) orthogonal polynnomials arc particular cases of Laguerre-Hahn orthogonal polynomials, and they satisfy a second-order linear differential equation.

The rth associated Laguerre-Hahn orthogonal polynomials are Laguerre-Hahn orthogonal polynomi-ais, therefore they satisfyil. fourth-order linear differential equation.

The search for thesc diffcrcntial equations has been very intensive during the past few ycars. For

r

=

1, Grosjean (1985, 1986) found them for Legendre and Jacobi families, and Ronveaux (1988). has given a single equation valid for tll(' first associated classical (continuous) orthogonal polynomials.

For an arbitrary r, computer algebra packages have been very useful due to the heavy computations involved. In this context we mention that Wimp (1987) has used the MACSYMA [ref] package to construct the fourth-order differential equations satisfied by the rth associated Jacobi polynomials (r in this case is integer or not). Belmehdi and Ronveaux (1989) devised a REDUCE program in order to obtain these differential equations for the associated classical orthogonal polynomials of integer (and fixed) order r.

Differentiai equations valid for the nh associated Laguerre-Hahn orthogonal polynomials and for any integer r were given by Belmehdi et al. (1991) using the properties of the Stieltjes function of the associated functional (see [Magnus, 1984]. [Dzoumba, 1985]). Then, followed some papers giving, in a simple way, the single fourth-order differential equation for the associated classical orthogonal polynomials of any integer order r (see for instance [Ronveaux, 1991], [Zarzo et al., 1993], [Lewanowicz, 1995]).

As it was the case for the associated orthogonal polynomial of a continuous variable, many works have been done to give the fourth-order difference equation satisfied by the associated classical orthogonal polynomials of a discrete variable.

Atakishiyevet al. (1996) have derived the relation (already known for classical continuous orthogonal polynomials [Rom-eaux, 1988]) giving the link between the first associated classical discrete orthogonal polynomials and the starting polynomials. and used this relation to prove that the first associated of the classical discrete orthogonal polynomials are solutions of a fourth-order Iinear difference equation which can be factored as product of two second-order linear difference equations.

Using the explicit representation of the associated ~leixner polynomials (with the real association parameter r ~ 0) in terms of hypergeometric functions, Letessier et al.(1996) gave the fourth-order difference equation satisfied by the rth associated Meixner polynomials and deduced by an appropriate limit process the difference equation for the rth associated Charlier, Laguerre and Hermite polynomials. This equation, thanks to the computer algebra system ~IATHEMATICA[Wolfram, 1993] and the relation proved in [Atakishiyev et al., 1996] is given explicitly for the first associated of Charlier,~leixner,

Krawtchouk and Hahn polynomials [Rom'eaux et al.. 1998a].

The question one can ask is whether it is possible to give one fourth-order difference equation valid for the rth associated Laguerre-Hahn orthogonal polyno:nials including orthogonal polynomials of con-tinuous, discrete \-ariable and also q-polynomials? The answer is yes and the first part of this dissertation aimed at answering this question.

1.1.2

The non-linear difference equations

Here, wc consider that the polynomials {P,,}nEf\i. orthogonal with respect the semi-classical linear

func-tional12 is orthonormal((L, P"Pn ) = 1 \/n E ~. , thus, satisfying

wherean and bn are complex numlwrs with (ln

i=

O.

The coefficientsan and bn can be given explicitly for c1assical (continuous) orthogonal polynomials in

terms of the polynomials 0 and

l:>

appcaring in the Pearson differential equation,

fx

(r/lL)

=

'ljJL, satisfied by the linear functional 12 with respect to which {Pn}nEf'i is orthogonal (see for instance [~ikiforovet al., 1983] [Chihara, 19ï8], [Szego, 1939], [Lesky, 1985j, [Koepf et al., 1996] ... ).

These coefficients are also known for classical orthogonal polynomials of a discrete variable and for q-classical orthogonal polynomials ([l\ïkiforov et al.. 1991], :Szego, 1939], [Lesky, 1985], [Koepf et al., 1996],[~ledem, 1996] ... ).

When the polynomials are semi-classical (instead of c1assical), except for some particular cases, it is difficult to give, in general situation, the coefficients an and b_{n .}

The propertics o'; the coefficientsan and bn as \Vell as those of the polynom;-als Pn have been

inves-tigated by many authors .

• Firstly, we cite for example Laguerre, who, in 1885, explored the properties of the orthogonal polynomials related to the wcight function p satisfying

p'(X)

=

R(

(

X; ,

J.1.

Historical introduction

9

whereR(x)is a rational function ofx. He also studied Padé approximations and continued fraction expansions of functions satisfying a differential equation of the form

W(x)J'(x)

=

2V(x)f(x)

+

U(x),

where U, V and1V are polynomials; and recoyered orthogonal polynomials Pn as denominators of

the approximants of

f.

He succeeded in shO\ying that the orthogonal polynomials Pn satisfy the

remarkable differential equation,

WGn y"

+

[(217

+

W')Gn - RTG~] y' -1-K_n y = 0,

where 0n andKn are polynomials with bounded degrees, whose coefficients are solutions of certain

(usnally) non-linear equations which provide non-linear equations foran andbn (see [Magnus, 1991]

for more details about Laguerre equations) .

• Secondly, we cite the works by Freud (see [Freud, 1976, 1977, 1986]) who investigated the asymptotic behaviour of the recurrence coefficients for special families of measures by a technique producing an infinite system of (usually non-linear) equations (called Freud equations) for these coefficients (see [Magnus, 1991] for more details about Freud equations). For example .. if the polynomialsPn

are related to the weight p(x)

=

exp(_x4_{) on the whole real line, then the Freud equations are} reduced to [Nevai, 1983]

{

-1a;'(a;'+1

+

a;'

+

a;'_I) = n, n

~

2, ao = 0,

aî

=

HH:l,

bn = 0, n ~O.

It should be noted that other people found similar non-linear equations and identities (see for instance [Laguerre, 1885], [Perron, 1929], [Shohat, 1939], see also [l\evai et al., 1986], [Nlagnus, 1991] for more details), but these authors did not study their solutions when no simple form cauld be found.

C sing the Freud equations, Freud (1976) gave a conjecture about the asymptotic behaviour of recur-rence coefficients when the polynomials Pn are rclated to the weight function

p(.7:) =

Ixle

exp(

-Ixia)

stating that :

Let an and bn be the coefficients of the following recurrence relation

satisficd by the polynolllials {Pn}nEN, orthogonal with respect to the weight p(x) =

I.rl e

exp(

-Ixia),

( >

~1, n

>

O. on the whole reallinc. Thenan and b" obcy:

Q 2r(a)

,,!~~ [n/C(~)p/O'

= 1, C(a) = f(a/2)2'

Important investigations have been devoted to the proof of Freud conjecture as weil as to the study of the asymptotics for

{P

n } nE"" the distribution of zeros, the sharp estimates of the extreme zeros ... ([Chihara,

1978], [Freud, 1976, 1977, 1986], [Lubinsky, 1984, 1985a, 1985b], [Lubinsky et al. 1986,1988] , [Magnus, 1984. 1985a, 1985b, 1986], [Bonan, 1984], [Matô et al., 1985], [Mhaskar et al., 1984a, 1984b], [Nevai, 1973, 1983, 1984a, 1984b, 1985, 1986], [Sheen, 1984] ... , for more details see [Magnus, 1984, 1985a. 1985b, 1986]).

Later, Belmehdi and Ronveaux (1994) gave a systematic way to obtain non-linear equations for the recnrrence coefficients, valid for any scmi-cla.ssical orthogonal polynomial of a continuous variable. In fact, given a semi-classicallinear functional L satisfying ddr.(cPL)

=

'lj;L, where cP and 'lj; are polynomials, they were able to provide two non-linear equations for the coefficients an, bn of the recurrence relation

satisfied by the polynomials {Pn}nEN associated to L, called Laguerre-Freud equations (denomination borrowed from Magnus [Magnus 1985b, 1986]).

In the second part of this dissertation, we givc a gcneralisation of the previous results [Belmehdi et al.. 1994] by giving the systcm of two non-linear differcncc equations satisfied by the recurrence coefficients: equations which are valid for semi-classical orthogonalpol~'nomialsof a continuolls and discrete variable. and also for q-semi-classical orthogonal polynomials (both of class 1).

1.2

Summary of the maIn results

1.2.1

The fourth-order difference equation

1. Using the l'l'suit in [8uslov, 1989], we prove the following:

Consider12a regular !inear functional satisfyingVq(1JL)

=

'l/JL, where 'l/J is a first degree polynomial

and 1>a polynomial of degree at most two. VI is the Hahn operator defined by

_ f(qx) - f(x) --J- . _ ,

Vqf(x) - ( ) , x -r0, q

i-

0, q

i-

1,Vqf(O) .- f (0).

q - 1 x

Then, if {Pn}nEN is the monic family of polynomials, orthogonal with respect to L, then, the first associated PAl) of Pnsatisfies the fourth-order difference equation

O" Qi,n-I

[Pn(~I(X;q)]

=0. -2,"-1 q2 (q _ 1)2x2

Operators Qi~n-I and Q;,n-I are gi\'en by:

with

Q;,n-I O"_~2.n-1

1>(2)

Q; -

((1

+

q)9 1)

+

'l/J(l) tl - .\n,Oti)Çq

+

q (0

+

1/J t) 'Id,

(1)(3)

+

'l/J(3) t 3)[q2AI

+

(1

+

q)1>(2)

+

1/J(2) t2JQ; _[q3

ih

(lh"2)

+

'l/J("2 t 2)

+

A3(1)(2)

+

qAI)]Qq

+q1>(l) [q2.4. 2

+

(1

+

q)1>(3) _{+'l/J(3) t 3)] 'Id,}

.\n,O -[n]q{'l/J'

+

[n - 1]1 - } .1>" [n]q

= - - .

qn - 1 q

i-

1, n:::: 0, QqP(x)

=

P(qx) VP EIP',

q 2q q - 1

ç(qi x ), 'l/J(i)

==

1.'(qi X), t,

==

t(q'.r), t(x) = (q -1)x,

(1

+

q)1>(j)

+

'l/J(j) tj - .\",0

fJ.

This l'l'suit [Foupouagnigni et aL, 1998d], i:, used to deduce the factored form of the difference cquatiolls satisficd by the first associated classical orthogonal polynomials of a discrete variable [Rom'eaux et aL, 1998a] and also the factore!: fonn of the differential equation satisfied by the first associated classical continuous orthogonal polynomials [Ronveaux. 1988], \Vc have used, also, this l'l'suit to prove that under certain condition:, on the parameters. the first associated of littl(' and big q-Jacobi polynornials arc still classica1. \loreover, we deduce that ifPn(x;a,bIq) (respectively

P"(x; a,b,c;q)) denotes the monic /ittle q-Jacobi polynomials (respecti\'ely monic big q-Jacobi polynomials), t!ten they are related \\'Ïth their respective first associated by:

( l ) . 1 p" (x,a. - Iq) qa ( 1 ) . 1 . PTt (x,a, - , c,q) qa, " n ( x l i ) a q P" - ; -, aq q , aq a n (x 1 a Pn - ; - ,aq, cq; q). a a

2. We prove t!tat tlil, rth ass(Jciated Vq-Laguerre-Hahn orthogonal polynomiaL satisfy the single fourth-order differencc cquatioll [Foupouagnigni et aL, 1998e]

~

L

Ij(n,r,q,x)V~PAT)

=

0, J=O

1.3.

Outline of dissertation

11

We use suitable change of variable and limit processes to extend the ab ove result to the rth asso-ciated Laguerre-Hahn orthogonal polynomials of a continuous and a discrete variable, respectively [Foupouagnigni et al 1998b].

\Ve apply this result to compute explicitly the coefficientsIj(n,r,q,x) for the rth associated classical orthogonal polynomials (including classical continuous, classical discrete and q-classical polynomi-aIs) [Foupouagnigni et aL, 1998b, 1998c, 1998e].

1.2.2

The non-linear recurrence equations

We prove the following theorem (see 8.1) which is the main result of the second part of this Dissertation.

Theorem

The coefficients f3n and 1n of the three-term recurrence relation

satisfied by the Dq-semi-classical monic orthogonal polynomials of class at most one,

{P

n}nEl\!, can be

computed recursively from the two non-linear equations

{

('th

+

[2n]~~)hn +1n+d

=

F1(q;f30, ... ,f3n;11"",'n), ('th

+

[2n

+

1]~ ~),8n+ 1~ln+1 = F2(q; f30.· .. , f3n;Il,'''' In+d·

3 2

4>j and

1/:

j are the coefficients of the polynomials rjJ and 'ljJ (4)(x) =

I:

4>jXj, 'ljJ(x)

=

I:

'ljJjXj)

j=O j=O

appearing in the Dq-Pearson equation, Dq(rjJL) = 1/)L, satisfied by the regular linear functional L. FI is a polynomial of2n

+

1 variables and of degree 2; and F2 a polynomial of 2n

+

2 variables and of degree 3, \Vith the initial conditions

(L, x)

f30 = (L,1) ,1/)211 = -'ljJ(f3o).

\Ve use suitable change of variable and limit processes toextend the previous theorem to the D and .6.-semi-classical orthogonal polynomials of dass at most one [Foupouagnigni et al., 1998a]. \Ve then

gin~the Laguerre-Freud equations for the g('ncralised Charlier and generalised Meixner of class one and use these equations (numerical and symbolic computation with :\Iaple V Release 4) to give a conjecture about the asymptotic behaviour of the cocflicients(ln and -,1/ of the generalised Charlier and generalised

Meixner polynomials of class one:

Conjecture

The coefficients

Jn

and 1" of the three-term recurrencc relation satisfied by the monic generalised Meixner polynomials of class one obey:

and those of the three-terrn recurrence relation satisfied by the monic generalised Charlier polynomials of class one obey:

hm ((3" - n) = 0, lim (rn - Il) = O. n---+CXl TI---+CXJ

1.3

Outline of dissertation

In Chapter 2 wc give SOIllC results and definitiolls on orthogonal and associatcd orthogonal polynomials.

Chapter 3 gives sorne useful properties of the operators Aq,w and Dq,w and the proof of sorne charac-terisation theorerns for Dq,w-classical and Dq,w-serni-classical orthogonal polynornials; characcharac-terisation theorern which are valid (by lirnit processes) for the operators dd_x' Dq and b..

Chapter 4 is devoted to the study of the Dq,w-Riccati difference equation satisfied by the Stieltjes function of the gi\'en associated linear functional. In particular, we prove that the affine Dq,w-Laguerre-Hahn orthogonal polynornials are the Dq,w-semi-classical orthogonal polynomials and conversely. In this chapter, it is also proved that the Dq,w-Laguerre-Hahn orthogonal polynornials can be obtained frorn the Dq-Laguerre-Hahn orthogonal polynornials by a change of variable.

In Chapter 5 we give the factored forrn of the fourth-order difference equation satisfied by the first associated Dq-classical orthogonal polynornials and \\2deduce the difference equation for classical

orthog-onal polynornials of continuous and of discrete variable. 'Ve also consider the situations for which the first associated of the little and big q-Jacobi polynomials are still classical.

Chapter 6 describes the rnethod used to obtain, for the general situation, the single fourth-order difference equation satisfied by the rth associated D, Dq and b.-Laguerre-Hahn orthogonal polynornials. The coefficients of the fourth-order difference equation for classical situations are also given explicitly.

Chapter 1 gi\'es useful coefficients for classical orthogonal polynornials like f3n, ln, Tn,l and T n,2' Chapter S presents the rnethod used to obtain the two non-linear equations for the coefficients of the TTRR satisfied by the Dq-serni-classical orthogonal polynornials of class at rnost one. We also show how these equations can be used to obtain the two non-lînear equations for the coefficients of the TTRR satisfied br the D and b.-serni-classical orthogonal polynornials of class at rnost one. The conjecture about the asyrnptotic behaviour of the coefficients of the TTRR satisfied by the generalised Charlier and the generalised Meixner polynornials of class one (conjecture obtained thanks to the two-non-linear equations) is also given.

The appendices l, II andIIIcontain the data for classical orthogonal polynornials as weil as the results on the fourth-order difference equations for classical situations.

Itshould be mentioned that:

• Chapter 2, devoted to the prclirninaries. is based on [Chihara, 1978]. [Guerfi, 1988], [Belrnehdi, 1990a], [Salto, 1995] and [Medern, 1996].

• ChaptE'rs 3 and 4 generalîse to the operator Dq,w certain results gi\'en in the above rnentioned rcferences.

Chapter 2

Preliminaries

2.0.1

The notion of topology

vVe recall the notion of wpology on polynomials and linear functional vector spaces. These notions have been defined in [Trèves. 1967], [Maroni, 1985. 1988], [Guerfi, 1988] and [Belmehdi, 1990a]. For these prelirninaries, we shall exploit the \\'orks by Maroni [l\Iaroni, 1988], Guerfi [Guerfi, 1988] and [Belmehdi, 1990a].

Let lP' be a vector space of polynomials in one real variable with complex coefficients, endowed with the strict inductive limit topology of the spaces lP'n. lP'n C lP' is the vector space of polynomials of degree

at most n. It satisfies _CXJ

lP'nClP'n-t-t, n~O, lP'= UlP'n,

n=O

-and is endO\\-ed with its na,ural topology which makes it a Banach space.

Let lP" be the dual ofII'. equiped with its topology which is defined by the system of semi-norms:

Il.e11,,

= sup

\.LÎhl,

k<n

wherc Nh dcnotes the mor:lents of the fllnctional

.e

with respect to the sequence {x"

ln:

_{M k = (.e)k}

=

(.e,x k). lP' and P' are Fréchet spaces.

,·re

consider'V the \·ect,.)r spacc gencrated by the elements {(~~rVnS}Il (D

==

/x) with its inductive

limit topology. 15 dmotes tI1C Dirac measurc: (15,J)

=

1(0), 1 ECCXJ(IR).

L(~t F be the lincar application:

---+ ---+ n F(d)

=

L

djxJ_. J=O (2.1 )

F verifies the following propert ies:

i) Fis an isomorphism defined on'V into lP'.

ii) The transpose tF of F, is an isornorphism defined on lP" into'V'.

iii) tF =F on lP". Thlls,

(F(L),d)

=

(.e,F(d)),Y.e E lP", \/dE 'V. (2.2)

Sinœ {( -~r D"J}n brms a basis of2" [:\1aroni, 1988], that is, any clement

.e

ofP'can be expressed

as

(2.3)

it follows that

F(L) = I)L)nxn.

n2:0

'V' is therefore the vector space of formaI series.

(2.4)

Remark 2.1 Let L(lP',Ir) (respectively L(lP", lP")) be the vector space of continuaus linear applica-tions defined on lP' intaIF' (respectively on lP" inta lp"). The transpose of any element of L(lP',lP') is an element of L(lP",lP"). We shall use this process to define certain elements of L(lP",lP") basically by transpasing thase ofL(lP',lp').

2.0.2

Notations

We understand by linear functional any element L oflp" and denote by (L, P) the action of L E lp" on

P E lp'. We also denote bl' IR the field of real numbers, iC the field of complex numbers and by N the set of integers. Henceforth, we will use interchangeably deg(<;6) and deg <;6 to denote the degree of the polynomial <;6. The operator D represents the usual derivative operator (V = d~) while the Kronecker symbol 6n,j is defined by { 1 if 6n ,j

=

0 if n = ) ,

ni-

j

2.1

Orthogonality and quasi-orthogonality

2.1.1

Orthogonal polynomials

Definition 2.1 A set of polynomials {Pn}nEN is said to be an orthogonal polynomial sequence (O?S) associated to the linear functianal L E lp" if

V nE N,

V m,n E N, m

i-

n,

V nE N.

(2.5)

Definition 2.2 A polynomial P is said ta be monie if its leading coefficient is equal to one (P

=

xn

₊

bnxn-1

+ .. .);

and a manic polynomial family is a one in whieh any element is monie.

Definition 2.3 A linear f7Lnctianal L Elp" is said ta be regular if there exists an O?S assoeiated to L. Remark 2.2 We state the fallowing praperties.

1. If L is a reg7Llar linear f7Lnctional, then there exists a unique monie (O?S) assoeiated taL.

2. If {P"}"EN is orthogonalwith respect to L, then {P"}nEN forms a basis oflp'.

3. Any polynomial family {?n}nEN with deg( Pn )= n Vn E~ forms a basis oflp'.

Remark 2.3 If {Pn}nEf.O is a set of palynomials with deg(P,,)

=

n \:In E N and L a given linear f7Lnctional then the follawing prap,'Tties are eq7Livalent:

i) (L, P,,?m)

=

0 ii)

(.c,

xm?n) = 0

\:Im,n EN, n i m and (L, PnPn )

i-

0 \:In EN. \:Irlî,n EN, O:S: m

<

n and (L,xnpn )

i-

0 Vn EN.

The following theorem, prO\'cd in [Chihara, 1978], givcs a necessary and sufficient condition for the regularitl' of a givcn lincar functional.

2.1. Ortbogonality and quasi-ortbogonality

15

Theorem 2.1 (Chihara, 1978) Let L be a linear functional and Mn the moment of order n of L defined by Mn = (L,X n ).

A necessary and sufficient condition for the existence of an orthogonal polynomial sequence for L is

~n

i

0 'tin E N, where the determinant ~n is defined by

Definition 2.4 (Chihara, 1978) A linear functional L is called positive-definite if

(L, lT(x))

>

0 for every polynomiallT that is not identically zero and is non-negative for all real x.

Theorem 2.2 (Chihara, 1978) The linear functional L is positive-definite if and only if its moments are aU real and~n

>

0 'tin E N.

The following theorem, taken from [Belmehdi, 1990a] giyes in a more general situation sorne characteri-sations of a regular linear functiona1.

Theorem 2.3 (Maroni, 1987, Belmehdi, 1990a) Let L be any linear functional; then the following pmperties are equivalent:

i) The linear functional L is reg71lar.

ii) There exists a polynomia.l sequence {Pn}nEN (with deg(Pn ) = n 'tin E N) such that

iii) For any polynomial sequence {Q11}nE:; (with deg( Qn)

=

n 'tin EN),

Theorem 2.4 (Szego, 1939, BelITlehdi, 1990a) Given a regular linear functional L, the monic or-thogonal polynomials (O.P.) associated to Lare given by

(L, QoQo) (L, QOQl) (C,QOQn-l) (C, QoQn)

1 (C, QIQO) (L,QIQl) (C,QIQn-l) (C, QIQn)

Pn(x)

=

~ (2.6)

n-l

(C, Qn-l Qo) (L,Qn-lQl) (C,Qn-lQn-l) (C, Qn-l Q,,)

Qo QI Qn-l Q"

where {Qn}nE~ is any manie polynomial family (with deg(Qn) = n 'tinE N); and,

2.1.2

Quasi-orthogonal polynomials

The notion of quasi-orthogonal polynomials was introduced in [Riesz. 1923] and extended by Maroni and Van Rossum (for more information see [Belmehdi, 1990a]).

Definition 2.5 (Belmehdi, 1990a) Let L be any linear functional and {Pn}nEN a polynomial family with_{deg(Pn )}

=

n \in E N. {Pn}nEf, is said to be quasi-orthogonal of order s with respect to L if

(2.7)

{Pn}nEN is said to be strictly quasi-orthogonal with respect to L if

(2.8)

Remark 2.4 (Belmehdi, 1990a) 1. Conditions (2.7) are equi valent to

while (2.8) is equivalent to

"vt~ 1,

"dt ~ 1,

(2.9)

(2.10)

2. ft follows from the definition 2.5 that if{Pn}nEN is orthogonal with respect to L, then {Pn}nEN is

strictly quasi-orthogonal of classs

=

0 with respect to L (see also /Shohat, 1937)).

3. Notice that quasi-orthogonality of class s = 1 was investigated in [Dickinson, 1961) and that the definition 2.5 was al50 given in [Chihara, 1957) and [Rom'eaux, 1979) but without the second condition: :JmE N, (L,_{PmPm- s)}'F O.

2.1.3

Other definitions

Definition 2.6 Given a polynomial f E ? and a linear functional !2E P'. the product of f and L, fL,

is defined as

fL

U

L, P) (L,

f

P) \iPE :.

Gi\"cn

f

an element of

r,

the application L ----+

f

L belongs to L

(r'. r')

and is the transpose of the following element ofL(r,r): P---+

f

P.

Definition 2.7 Given a polynomial g E

r

and a linear functional LE IP", the product of Land g, Lg,

is a polynomial defined as where n n Lg(r)

=

L

L9dL

xk-J)x). j=Ok=j n g(x)

=

LgjXj. j=O (2.11)

Givcn a functional C, the application P ~ LP bclongs ta L(IP'.IP'). By transposition, wc define the product of two linear functirJllals Land.\If as:

2.1.

Orthogonality and quasi-orthogonality

Definition 2.8 The product of two linear functionals Land M is defined by (LM, P) = (L, MP), "iPElP'.

Definition 2.9 (Belmehdi, 1990a, Dini, 1988) The operator (Je is defined as

17

where c is a complex number.

= lP'-+lP' { P(x)-P(e) x-c ' P'(c), xi-c x=c (2.12)

The application Be belongs to L(lP',lP').

Definition 2.10 Consider the linear funetional L. From the above definition and by transposition (see remark 2.1), we define the linear functional (x - C)-IL, as

wherec EC.

(x - C)-IL

(x - c)-lL, P)

lP'-+C

(L, (JeP) "iF E lP', (2.13)

Corollary 2.1 (Belmehdi, 1990a) For any complex number c, and for any linear functional L the following holds:

(x - c)[(x - c)-l L]

=

L, (x - c)-l[(x - c)L]

=

L - (L, 1)6_{e ,} where 6e is the Dirac measure at the point c.

2.1.4

Dual basis

(2.14)

Definition 2.11 (Maroni, 1988) Let {Pn}nEN be a manie polynomial family w'ith deg(Pn )

=

n "in E P;. Then {Pn}nE'; forms a basis oflP' and ther'efore genemtes a. unique basis of lP", called dual basis

asso-ciated ta {P,,}n-':N, denoted by {Pn}nEN and satisfying

An~'clement L of f' can be expresscd in this basis as (sec [Roman et. al.. 1978], [Maroni, 1988]): L =

L

(L,Pn)Pn .

n~O

(2.15)

(2.16)

Proposition 2.1 Let L be a regular lincar funetional, {Pn}nEN the corresponding monic orthogonal

family and

{P

n }nE:--; _{the dual basis associatcd to {Pn}}nEN. We have

(2.17)

Pra0[: Let us write PfiL =

L

en,jPj. We obtain j

by the faet that {Pn}n--::N is orthogonal with respect to L. Thns,

2.2

Associated orthogonal polynomials

2.2.1

Three-term recurrence relation

We first give the following theorems which we shaH use further to define associated orthogonal poly-nomials. The first is taken from [Chihara, 1978] and the second from [Favard,1935] (see also [Wint-ner, 1929], [Stone, 19321 ,[Sherman, 1933], [Shohat, 1938], [Peron, 1957]).

Theorem 2.5 (Chihara,1978) Let L be a regular linear functional and {Pn}nEN the corresponding monic orthogonal polynomials. {Pn}nEN satisfy a three-term recurTénce relation

{

Pn+l (x) = (x - 13n)Pn(x) - '1nPn-l (x), Po(x)

=

1, PJCx) = x - 130,

where 3 n .md -'n are complex numbers with 'In i- 0 "inE N.

n

2:

1,

(2.18)

Praof: Since {Pn}nEN is orthogonal with respect to L, it forms a basis ofIP' (see Remark 2.2). We therefore expand the polynomial xPn on the basis {Pn}nEN and obtain

n-2

xPn= Pn+1

+

13n P n

+

'In Pn-1

+

L

Tln,j Pj, n

2:

1, j=O

(2.19)

where ln, 3n and Tln,j are complex numbers.

To compute 'fIn,j, we apply the linear functional L to both sides of the equation obtained after multiplying the previous one by Pj , j

:s

n - 2 to get

'fIn,j IO,j = (L,xPnPj )

=

0, j

<

n - 1,

with IJ,n = (1:-.PnPn ).

Cc.nsidering the fact that IO,n i- 0 "in E N (see (2.5)), it follows from the above equation that Tln.j

=

0, j

<

n - 1. Therefore equation (2.19) becomes

xPn

=

Pn+1

+

BnPn

+

ln Pn-1 . n

>

1.

:\Lmicking the approach used above to computc 'fIn.j, but \Vith the previous equation, we express 'In as

Rence ln i- 0 n :::: 1.

By convention one takes~io

=

(L, 1).

Ttc converse of the above theorem is due to Favard (1935) (see also [Chihara,1978]) .

D

Theorem 2.6 (Favard's Theorem) Let {13n}nEN and bn}nEf' be tlL'O sequences of complex numbers and let {Pn}n,,=~ be the family of polynomials defined by the recurrence formula

n

2:

1,

Then, there exists a unique linear functional L suclt that

(L,I)

=

'10 and(L:PnPm )

=

0 "im,nE N, ni- m.

L is ~gular and {Pn}nEN are the corresponding monic orthogonal polynomials if and only if 'In i-0 "inE N,

while L is positive-definite if and only if

2.2.

Associated orthogonal polynomials

2.2.2

The first associated orthogonal polynomials

19

Definition 2.12 Given a regular linear functional L and the corresponding monie orthogonal polynomials

{Pn}nEN, the first assoeiated of the polynomial Pn is a monie polynomial of degree n, denoted by PAl)

and defined by

p~l)(x)

=

~(L,

Pn+l(x) - Pn+l(t)) '<In E N,

ÎO x - t

withÎO = (L, 1). ft is understood that the linear funetionalL aets on the variable t.

I~eIllma 2.1 The monie polynomial family {PAl)}nEN satisfies the three-term reeurrence relation

{

P~~l

(x) = (x - 3n+dPA1) (x) -

~'n+1P~~1

(x), n

2:

1,

p~I)(X) = 1,pil)(x) = X - (JI,

where(Jn andÎn are defined in (2.18).

(2.20)

(2.21)

PraoE: Using the three-term recurrence relation satisfied by {Pn}nEN(see (2.18)) and (2.20) we obtain

= ~(L, Pn₊2(x) - _Pn - 2(t)) ÎO x - t ~(L, (x - (Jn+dPn-l(x) - În+1Pn(X) ÎO X - t (t -(Jn+dPn+l(t) -În+IPn(t)) x - t ( '" _ 0 )~(r Pn-I(x) - Pn-c-I(t)) :.1. Un-l L . -~!O X - t 1 P,,(x) - Pn(t) 1 , -În~1-(C ) - -(L,Pn+1(t)) ~!O X - t ~(o

(x - /3n-dPAI)(x) -

În+IP~~JX)

\fn EN.

o

We dcduce from Thcorem 2.6 and Lemma 2.1 that there exists a unique regular linear functional ÔI₎

with respect to which {PAl)}nEI'; is orthogonal with ([(1',1) = Î I '

Iterating the abovc process, we define the general associated orthogonal polynomials.

2.2.3

The rth associated orthogonal polynomials

Definition 2.13 Let [ be a regular linear functional and {P,,},,-=r; the corresponding manie orthogonal palynomials satisfying (2.18).

The rth assoeiated of the orthogonal polynomial P" is a polynomial of degree n, denoted PAr) and

defined by with p(r-l)() p(r-I)() P(r)( )

=

(r(r-I) ,,+1 X - "+1 t) Î r - I 71 X L , . x - t n

2:

0, r ~~ 1, (2.22) (Llr),1) = Îr> r

2:

1,

as.mming that ÎO

==

(C 1), P,~O)

==

P", and [l0)

==

L; u:here [Ir-I) is the regular linear functional with respect ta whieh {PAr-I)}71E~'is orthogonal; and it is understood that[lr-I) aets on the variable t.

Lemma 2.2 If L is a regular linear funetional and {Pn}nEN the eorresponding monie orthogonal poly-nomials, then, the rth assoeiated polynomials {PAr)}nEN of {Pn}nEN satisfy the three-term reeurrenee

relation

(2.23)

PraaE- We shaH prove the lemma by induction on r. For r = 1, (2.23) is satisfied thanks to Lemma 2.1. We suppose that (2.23) is satisfied up to a fixed integer r. Then using (2.22) we obtain

p(r+I)( ) n-I X p(r) ( ) (r) ( )

...!-(d

r ) , n+2

X~n+2

t ) Îr X - t

...!-(d

r ), (x -

f3n+r+dP~11

(x) - În+r+IPAr)(x) Î r X - t (t -

3n+r+dP~11

(t) - În+r+IPAr)(t)) x - t (X -f3l1+r+1)...!-(l'(r)L ,

p~11(x)-p~11(t))

t Îr

x-_~

...!-(d

rl PAr)(x) - PAr)(t)) _ ...!-(L(r) p(r) (t)) fn+r+1 , t ' n+1 Îr X - Î r ( f3 ) p (r+l) ( ) p(r+l) ( ) \..1 "'T X - n+r+1 n X - În+r+1 n-I X Vn E 1'1.

Thus{PAr)}nE"l satisfies (2.23) 'r:Ir E N. As consequence of the previous lemma, [Belmehdi. 199ÜbJ).

o

we daim the foHowing known result (see [Magnus, 1984],

Lemma 2.3 (Magnus, 1984, Belmehdi, 1990b) The aS.5oeiated polynomial.5 PAr) satisfy

n

P (r)p(r+l) _ p(r) p(r+l) -11 n n+1 n - I -

II

rr-k -

=

Irn,k

k=1

PraoE- In the first step we \\Tite (2.23) for P,;11 and p,;r+l)

'r:In E N, 'r:IT' EN. (2.24)

p~11(x)

p,~r+I)(x)

(2.25) (2.26) In the second step we subtract the two equations obtained after mllitiplying (2.25) and (2.26) by p~~~I) and pAr), respecti\'ely,

p(r)p(r+l) _ p(r) p(r+l)

=

,(p(r) p(r+l) _ p(r) p(r+I))

n n n+1 n-I În-r 11-1 n-I n 11-2 .

Then relation (2.2-1) results by iterating the latter.

2.3

Operators

D,

1:,

D

_{w ,} Qq

and

D

_q

2.3.1

Operator

D

o

The application P -+ 'OP belongs to L(II",?). I3y transposition, we define derivative of the linear functional as:

2.3. Operators

D,

T:,;,

Dw ,

9

q and Dq

Definition 2.14 Let 1: be a given linear functional, wc define the D-derivative of1:, DI:, as

21

DI:

(DL, P) -(C DP) VPE IF. (2.2ï)

Proposition 2.2 Let 1: be a regular linear functio nal, {Pn}nEN the eorresponding monie orthogonal

family and {Pn}nC; the dual basis assoe/atcd tu {Pn}nE::- If {Qn,dnEN is the dv,al basis assoc/ated to

the monie farnily {Qn,dne; defined by

o

= DPn - 1

,.n,11/.+1' then we have

DQn1 = -(n

+

1)Pn+1 •

Praof: This fo11ows from Proposition 3.5.

o

Definition 2.15 The regular lincar funchul/al 1: and the eorresponding monie 07·thogonal polynomials arc said to be D-semi-c/assical (or semi-c/a',ica! eontinvous) if there exist two polynomials

li'

of degree at lcast one, aud <P sl/,ch that

D(ol:)

=

1/'C (2.28)

Moreover, if<P is a polynomial of degrcc at most tu'o and Ji; a first-degree polynomial, then, the linear' functional and the eorresponding orthogonal polyno mials are cal/ed D-classical (classical eontinuous). For more details about D-scmi-c/assical orthogonal polynornials can be found in [Maroni, 1985, 1987},

[M arcel/àn, 1988}, [Belmehdi, 1990a} and references therein.

2.3.2

Class of the V-semi-classical linear functional

Let 1: be a D-scmi-classical linear functional satisfying

D(01:) = lI'L,

whcrc <Pis any lIon-zcro polynomial ;\lld (i, <1 polynomial uf degrcc al. kast one. L: satisfics

D(f<pI:)

=

((JDf

+

(Jf)C for any polynomia: f·

Definition 2.16 We r/ejillc the rias, d(1:) I)f thc D-sernl-c/o8s/collinefl7' functional 1: as

cl(l:) = mill

o {max(ckg(f - 2.dcg(g) -1)}, (J ..'}),=7",

where

RI

=

{(f ..IJ) EIF~ /deg(g)

2:

l (mdD(fI:) =gl:}.

(2.29)

Proposition 2.3 (Belmehdi, 1990a) If1: isa D-scmi-classicallinear fvnctional satisfyiny(2.29), then 1: is of dass s

=

mélx(dcg(0) - 2, dcg(1/J) - 1) if and only if

Il

1r

el

+

1(1:,1/',')1)

cl

0, (2.30)

where Zr/> is the set of zems of6. The comp/ex Tlumberr~ aud the polynomials <Pc, 1/Jc a7'e defincd by

Pro of: for a proüf sec Proposition 3.4.

(2.31)

o

Remark 2.5 It fol/mus fnmt lite definitiou of the c/ass !Jf the /iumr fuuctioua/ lhal tlu: D-dassiml /iuear funetional is a D-scmi-dassù,a/ li'T!l:IlT" fuucfllllwl of c/as., s

=

o.

Lemma 2.4 Let L be a regular linear funetional.

i) If there exist two polynomials 11)

i:

0, and 0 sueh that D(oL)

=

1/;L

then 9is a non-zero polynomial.

(2.32)

ii) Conversely, if there exist two polynomials

9

i:

0 and1/; sueh that (2.32) holds, then1/; is of degree at least one.

Proof: For a proof see Lemma 3.1.

2.3.3

Characterisation of V-classical orthogonal polynomials

o

The following t heorern which is a corollary of the theorem 3.1 gives sorne characterisations of classical con-tinuons orthogonal polynornials (see [Chihara, 1978~.[l\ikiforov et al., 1983], [Al-salam, 1990], [Marcellan et al.. 199"*], ... ).

Theorem 2.7 Let L be a regular linear fu/netional, {Pn}nEh the eOT'T'esponding monie orthogonal family and Qn,rn the monie polynomial of degree n defined by

with

(n

+

m)!

En.n>

=

Qn,O == P n. n!

The following properties are e'luivalent:

i) There exist two polynomials, Ô of l1eyr'(;(~ at rnost two and l' of degree one, sueh that

V OL

=

'le

ii) TliiTC c.rist two polynoTTuals. 0 of l1egl,(;(; nt Tnost tUiO and~' of degree one, sueh that for any integer rn,

vr

oC"')

= /.

·",.L.

(Lm.Q),,,Qlllf')

= /;

,i5).,.7).n Ef~, Ik" -j::.0\:/nEN),

with the linea.r funetional Lm and tlu' [lolIjTlI)TT/,ilJ.i ~"71 defined, reeurslcely, by

1.)",-1

=

Do

+

l'm, 00

==

1),

L'''_I

=

<pL",. Lo

==

L and given expl1eitly by

'l;'m(X)

=

mo'lx)

+

1/;(x),

Lm = <pm L.

lii) Then: exist two polYTlomiai." (J of degree at most two and '0 of degree one, such that

for (Lny integer' m, the foilol1!1'TtlJseumd-order differ'ence equation Iwlds:

with the polynomialljJrn given !;y (.!.:n) and the constant

>-;,,111

given oy

<pli <pif

>';,m

= -

TI { /.

<II "

{n - 1;""2 }

=

-Tt{'l}/

+

{21T!

+

Tt - 1)""2 }.

(2.33) (2.34)

2.3.

Operators

'D,

7.,;,

D

w , Ç/q

and V

q 23

iv) The1'e e:r;lst twu polynufTlials, 0 of degree at most two and 1/J of degree one, such that, for any

io'tc9P1'm, thc following relation halds:

(2,36)

v'ith the polynomial l'm, the linc(/7' functional

L,"

and the constant

>-;1.m

gil'en, respfctiveiy, by (2.3.1). (2.34) and (2.35).

v) There e.List a polynomial 0 of degree at most two and thrce constants Cl l .n +l, cn,n, Cn,n-l with

Cn . n-1

#

0 S1ICh that

vi) For any non-zero intege1'm, th ere exist a sequence of complex numbers {lin,m} nE1\i such that

Qn,1n-1 :=Qn.Tn - U n -l,17,Qn--l,Tu

+

vn -2,ntQn-'2,Hl' 't/nEN-{O,l},

Rernark 2.6 Let us comm.ent on the aboue pmpe1,ties.

For ailm E :~. the derivative of onlerm, {Qn.m}nE"l, of the family {Pn+m }nE'; is classical and orthogonal

with 1'CSPCet to the c!assical linmr functional Lm.

The functional version of the gen eralised Ro(h'igues formula [Nikiforo1J et al, 1983}, .Belmehdi, 1990c). given below, is obtained by itemting the relation (2.36):

11-1

Qn,1ll01Jl_L

=

II

_{1 " vn(On+m 12).} J==U ~J'

+

(j

+

2m

+

11 - 1) <Pz

2.3.4

Operators

~

and Du;

Definition 2.17 The arithmetic 5hift opcmtorT"" i5 defincd by

T.. . -

---t IP'

P ---t TwP, T""P(.r) = P(.r

+ :..:),

~. E lIt IVe dello/.e

0.

=

T.

Definition 2.18 The di1Terc!l.cc '7]JeFJfor D~, 15 defined by

(2.3ï P ---t lI" P(x+:.u) -Pt;];) _ D _ P, DwP(;c) = ,..u E :::, :.u

#

O. LA: (2 ..38

Wc deTlote Dl = 6. and D_1 = \ . ~ and V denote the fOT1L'ard rmd the backwo.rd difference opemtors. 1·e.9pecti I,e/y.

The applications P ---t TwP am: P ---t Dv..'P bclong to L(IF.lP). We, therefore. use their transposes to ddine the action of the operator::' Tv... aml Dw on the lincar fUIlctionals.

Definition 2.19 The action of thc arithmctic shift o[!cmtol'Te; on the f1Lnctional 12 is defined by

(~C P) = (L,T-wP) 'iP E lP'.

Definition 2.20 Given a Zil/car functional L, wc rle[inc the De; rlcrivativc ofC Dv..'L, as

(2.391

I?

-He

Definition 2.21 The regular linear fU71ctional [ and the corresponding monie orthogonal polynomials

are said to be Dw-serni-classical if the1'c exist tu'o polynomials

l/J

of degree at least one, and 0 sueh that

D~(9L) = 1,.-[, (2.-11)

Moreover, ifif; is a polynomial of degnl" at most tu'o a71d~' a first-degree polynomial, thcn, the linerlT' functional and thc corrc.spondlng O1't1wgonal polyno'llllal.s are ealled classical dis crete.

Using the abo\'e definitions. \\'e obtain the follo\\'ing properties: Proposition 2.4 (Salto, 1995) T_",D_

Tu.

(f9 D~,(fg D~lf[ D wT_~ = D __" DwD_~, = D_.D",.. T",f~g,

T.oU [)

= ~fT..L f Dk·y-t- ~gD~f =

T..

f

D.g

+

9D~I f D k·[+ D",f

Tu..[

=

T.fD~,[- D_fL.

lTu. -

'Id)C

~,fDAflL)-

T..

f D. 9[ +D.Ug)L. "11,9 EC. 'iL E If',

(2.-1~) (2.,,[,3) (2.-1-1) (2.-1:;) (2.-16 ) (2.-1';")

?\orice that equation (2.42) means that:

T-wDw4>

LT_wDw<I-

==

D~,T__ 4>

=

D_",4>. D",D_~if>

=

D __.D",if>, Vif>E ?, D~T_.'Î>= D_",'Î>. D",D __ <f> = D __D~,<f>,

V<f>

E ?'.

Pro of: This follows direetly from Proposi~ion3.1. ::::J

The following lemma prows that th,' arithmetic shift of the assoeiated orthogonal polynomials (resp, regular linear fune1 ional) are the <lSSOI iatcc! sLiftcc: ort hogonal polynomials and shifted regular linear funetional, rcspcctivcl:-',

Lemma 2.5 GiveT/. a "eflu1m'!inco!' fUT r-tiOTwl ~ an; {I"n

}"E

the cOTTC.lponding T1l'Jnic orthogonal pollJ-1l0rnials, the rth associrzted p,'1 of P" :ndJ2{1, of [ obel)

'-: r, n ::: Ii. (2.4~)

Proof: \\'(' shall ;in' tL,' plOuf ])\ indllC1: ln 0:. r. It fo::ows fro::1 L(llll1k. ;)2 that

{T..

P,,},",: ;,re

the monie onhlJ12,onal polyno:ni<l1.s a,oo iiltcd te T..~,

for l' = (J 1Tu.:F,,( 1=

T..

r/,O)

=

T.

l)" alJ(i (T... ~)(O = T, [1"1 =

T..,

C

Suppo:"e that (2.~t") is sari,;fÏi'd 1lp ,') ;\ fix.c1 r, Then llc:ng 2.2:':) and tll'~ :ar r Thar [ acts on r;jP

\'ariable t, \\le ~et

TLen, T P (r+:'. l '" J)) 1J) = (T f ) )(r'( ( P .,) ~«(T.

[f

.

l ,,-1 X - T", ,+) ) x - t

_1..

fT [(r),

'G

F~~1

(x) -

T..

Pr;';,1

it) -'r ' " - . x - t T p(e (x1 _ p(r) ft) .2.(T Llr ),T, -' , , - 1 . ,,~1 ) -, r , - . ~ x - (t - _.) (-, ( . p(rl ) ..1..(['r ,[>,,-1 X

+-v' -

,,-1 t ) -'r .r

+ _. -

t T_. F//-1)(I 711 E N, 71' E N, (2A9)

2.3.

Operators D,

Tw,

D

w , Çiq and Dq

\V€~ use remark 2.3 to get

For n = 0 (see dcfinition 2.13),

V;e combine (2.49), (2.50) and (2.51) to get

25

(2.50)

(2.51)

Renee (Twq(r+l)

=

Tw L(r+l), thanks to the fact that

{'J...

Pr;C+l)}nEN, which is orthogonal with respect

to

'r.,

L(r+l), forms a basis of]p'. 0

2.3.5

Class of the D",-semi-classical linear functional

Let L be a Dw-semi-classicallinear functional satisfying

(2.52) where 0 is any non-zero polynomial and"l/J a polynomial of degree at least one. L satisfies

D_ Urj;L) = (rjJDwf

+

"l/JT;.;f)L, for any polynomial f·

Definition 2.22 Wc define the classcl(L) of the D",.-semi-classical linear fu,nclional L as

cl(L)

=

min {max(deg(J) - 2.deg(g) - 1)},

(f.g)E'R2 where

R2 = {(f,g) E]p'2 /deg(y)

2:

1 and DwUL) = gL}.

The following proposition giYC a characterisiltion of the class of a Dq-semi-classical Enear functional.

Proposition 2.5 (Salto, 1995) If L is a D~-81:mi-classiCll/ linear' fuuctiorwl .'atisfying (2,52), thea L

18 of da8S:; = lllax(deg(ç~)- 2,deg(l.') -1) If Ilnd on/y if

(2.53)

where Zq, i8 the set of zcros ofrjJ. Thc camp/ex uurn/)C1' rc .w and the polynomials rjJc. Llc.,,- are dcfined

uy

(x - c)rjJc =

1),

l ' -

9c

= (x

+

w - c)~'cw

+

r

c.

w· (2.5-l)

ProuE: This follows froIll Proposition 3.4. D

More details about the class of a De.;-semi-classicallinear functional can be found in [Salto, 1996] and [Godoy et al., 1997b].

Remark 2.7 From the definition of the cla58 of the scmi-classical linear functional. we deduce that the

D _. -classical linear flluctioual is a D..J -8r:mi-r:fa881àll linear fu n'clional of class s

=

o.

Lemnla 2.6 Thc lincljr fuuctional L 18 n:qlliur if aud only if TwL is refJular.

ProoE: For a proof see LClllIlla 3.2,

i) If thel'e exist two polynomials 'l' :j:0 and4) su ch that

D.J(<jJL) = 1jJL, (2 ..:55)

then 4) is a non-zero polynomiol.

ii) Convcrsely, if there exist two polynomials 0 :j: 0 and

1/)

such that (2.55) holds, then 7j; is of dcgree

at lcast one.

Pra of: This follows from Lemma 3.1.

o

Proposition 2.6 (Salto, 1995) Let L be a reglllm' linear functional, {P,,}nEN the eorresponding manie orthogonal family and {Pn}",,:ci the dual basis assoeiated ta {Pn }"EN· If {Qn, dnECi is th e dual basis

assoeiated to the monie family {Q11,dnE~ defined by

{) _ D_.P1l-1

- 11,1 - n

+

1 '

then we have

Praof for a proof see Proposition 3.5.

2.3.6

Characterisation of

~-classical

orthogonal polynomials

o

The following theorem which is a corollary of Theorem 3.1 gives a characterisation of the orthogonal polynomials of a discrete variable [Al-salam. 1990], [~ikiforov et al., 1991], [Garcia et al., 1995], [Salto, 1995].

Theoren12.8 Let L be a regidal' lincal' funetional, {Pn}"Ci the cOITesponding manie ol,thogonal family and Q",11/ the monie polynomlll.l of Jeun:!' Tl defined by

2.56) with Dn,lll == (71 +m)' -n!

0",0 ==

P". 2.5ï)

The follo wiufj pmpcl't1cs arc, 'jll.ivall ni.

1) Th/TC (ri.~t two ]!ulynorllial.'i, (J of dCfjJ'cc (Jt must tu.'O and l ' of degree one, sucli Ihul 6.(oL:) = 1/:[,

ii) Then: exist twu polYTlollàais. 0 of dCfjrcc at most tl1'O and1.1' of dCU1Te onc, sll.chtllfLffor Il,ny mtcuer

111.,

6.(OL",) = ('JI1/L.

(L",.Qj,,,,On.,,,)

=

k"oj.", (k,,:j: 0\:/71 E N),

with the linear fu nctional Lni and th e ]!olyn omial1jJm defined, recursive1y, by

1/Jm- l = .:l<jJ~TL'_{m ,} 1/)0

==

1jJ,

Lm-l = T(rjJ Lm)' LO

==

L

and UiTJcn CXIJlicitly by

C'''L(:C) = rjJ(x

+

m) - o(x)

+

V··(x

+

rn), n. L"L

=

Il

(/i(x

+

j )

rH [,

J=1 (2.58) (2.59 )

27

iii) There exist two polynomials, 0 of rlegree a t most two and~' of degree one, such that for any integerm, the following second-arder difJerence Equation holds:

with the polynomial 'I,)m given by (2.58) and the constant Àn,m given by

9"

</J"

À;,

m

= -n

{1/';n

+ (n - 1)~} = -n

N'

+ (2m + n - 1)-}.

, 2 2 (2.60)

Iv) There exist two polynomials,

9

of degree at most two and 1/J of degree one, such that, for any integer m, the following relation hoMs:

(2.61)

with the polynomial1j;n" the linfar funct/ona! Lr ; and the constant À~.m given, respective!y, by (2.58),

(2.59) and (2.60).

v) There exist a polynomial 0 of degrct at rnost two and three constants Cn .n +l. C_{n ."·} C_n,n-l with

Cn,n-l -::j0 such that

vi) For any non-zero integer m. there exist sC'luence of complex numbers {un,m}nE', such that Qn.m-l

=

Qn,m

+

111l~I,mQll-1,r"

+

1·n-2.mQn-2,m, "In EN - {O, 1}. (2.62)

Remark 2.8 1. For all mEN, the ~-derivativeof order m, {Qn.m}nE"", of the family {P',-l-m}nEN

is classical discrete and orthogonal with respect to the classicallinear functional Lm.

2. The analogue of the functional vers ion of the generalised Rodrigues formula (Nikiforov et al .. 1991),

(Salto, 1995)given below, is obtained by itemting the relation (2.61)

nt n - l 1 n~7n

Qn,1/I

II

</J(:r

+

.i)T'''L~ =

I I , ,

6" \ n (

II

</J(X

+

j)T"+rrtL).

j=1 FO L' + (2m +] +n - 1 ' 2 ]=1

S. If the linear functiona! L /05 T'é]lT'esentu! by the positive weightp on the lntervall

=

:'1 .

br, (C p

=

~ pl.r)P(.T) VI' Ef,

~

0"'ôJ

11'ith xn_</J(x)p(x)l~

₌

_{0 7/1 E IJ, ther! we have the EquivalEnce}

2.3.7

Operators

QI[

and

D~.

Definition 2.23 The gco'//!dric shijt oJ!f:mtor

ç;

105 definerl by

- - t

- - t

(2.63)

(2.64)

(2.65)

Definition 2.24 (Hahn, 1948) The q-rlifference operator Dq • called Hahnoperator 1s rlefined by

l'(qx)-P(x) _

D,/I',D,/I'x)= ( .rJE_-'.:,q-::jO,qil.

q - l)x (2.66)

The applications

r

- - t (i,Il' il1ld l' ----+ DqP helon~1.0L(?,?). \Ve, therdore, lt;;(~ IL·'ir transposes 1.0 dcfÎlw the action ofthl' Op(~rilt(Jr~Ç'I alld Dq 'Jn the lim'ar funl'tionals.